包含结构归纳原理的形式数学系统
Formal Mathematical Systems including a Structural Induction Principle
摘要：
我们提出了一个统一的理论，正式的数学系统，包括递归系统密切相关的正式文法，包括谓词演算以及正式的归纳原则。我们引入递归系统产生递归的术语列表之间的关系，考虑的基本对象。递归系统由公理和特定的推理规则组成，公理是一种特殊的无量词正角公式。它的扩展到正式的数学系统，导致一个正式的结构归纳的基本递归系统的公理。这种方法提供了一些新的表现定理，而不使用人工和困难的解释技术。在这个框架内，我们还将推导出哥德尔第一和第二不完全性定理的一般类公理化的形式数学系统的版本。
英文摘要：
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating the recursively enumerable relations between lists of terms, the basic objects under consideration. A recursive system consists of axioms, which are special quantifier-free positive horn formulas, and of specific rules of inference. Its extension to formal mathematical systems leads to a formal structural induction with respect to the axioms of the underlying recursive system. This approach provides some new representation theorems without using artificial and difficult interpretation techniques. Within this frame we will also derive versions of Gödel's First and Second Incompleteness Theorems for a general class of axiomatized formal mathematical systems.
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